Intrinsic coordinates in school mathematics

Intrinsic coordinates are included in the A level Further Mathematics syllabus followed at my school. In the text books produced specifically for our examination board, intrinsic coordinates are introduced via a diagram similar to the one given here from Wikipedia.

Our book then says `From some fixed point of the curve the length of the arc is and the angle often called the gradient angle, is the angle made by the tangent to the curve at with the positive direction of the -axis.It is clear that for every point on the curve, and continuously change and so do and ’.

My immediate questions are the following:

Why is it OK to talk about `the’ angle?Which angle to we mean if the function is decreasing?If we mean the obtuse angle, how can vary continuously as moves over a minimum?If is to vary continuously, it must pass through 0 and become negative, so is it that is defined by = tan ? If we do take this definition for why is there no mention of the fact that can be negative?

I consulted various other books and my colleagues, and looked for something helpful on the web, but made no progress.The diagram and explanation above appear to be the standard ones, but do not help me (or my students) to answer to my questions.The most comforting response I received was from a retired teacher who told me that I was right to be thinking about it, but that the exam questions would avoid any potential difficulties.

I should like to know your thoughts on these questions and about the value of introducing this topic so informally this at this stage in a student’s education. In only a few months these students will be encountering real analysis and abstract algebra, and be expected to construct formal proofs. How does this style of `pure’ maths prepare them for this?

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Ah, Edexcel FP2. Let me start by saying that this is a topic widely denounced by Edexcel FM teachers, largely because it (and indeed the entire coordinate geometry chapter) is, effectively, nothing more than an exercise in using the formula book.

Indeed, if the function is decreasing, the angle should be negative, in the “usual” anticlockwise-from-rtight convention. Actually (I am almost sure) the curves studied need not be graphs of functions of x; the curve can proceed from right to left as well as left to right. I agree that this is potentially problematic with respect to the continuity of phi. And yes, thinking of phi as arctan dy/dx is probably your best recourse.

I’ve already touched on the value of this topic: zero. I had hoped that when they rejigged the further pure courses this summer it would go, but it isn’t.

You ask: “In only a few months these students will be encountering real analysis and abstract algebra, and be expected to construct formal proofs. How does this style of `pure’ maths prepare them for this?”

Badly. Negligibly. Criminally.

The one justification, I suppose, is that further maths is not (and neither should it be) a course only for future undergraduate mathematicians, any more than A level physics should be taken only by those who will study it at degree level.

The intrinsic coordinates are not global, they are local. The alngle itself is only locally a number too, although it can be viewed as a number mod . These to points taken into account, why not use them?

Thank you for your message dr rick. It is a relief to hear that there are other teachers out there who have a problem with the motivation for exam board’s approach to this topic. I agree that further maths should not only be a course taken by future undergraduate mathematicians, but I do not feel that the questions I was asking should be any less natural to a potential physicist or engineer. It also seems dangerous to encourage students to use phrases like `clearly varies continuously’. Undergraduates generally resented my circling phrases like this in their real analysis work.

I also expected the topic to vanish from Alevel (it was required for M6, but that is long gone). Is it definitely in the new spec? I have had a quick look, but I can’t find it.

To misha, good point. It is not the first time students have had to deal with a mod definition in further maths. I have found this topic difficult to teach partly because I am not particularly familliar with it and partly because local resources were limited – both of these aspects interested AB, which was why he asked me to post the problem.

EHK – I discovered today (to my vast delight) that, while most of the coordinate geometry chapter survives (in the new FP1) the intrinsic coordinates are gone. Apologies for unwittingly misleading you. We have six people in our department who teach further pure, and they all hate that topic.

Now all we need is some decent linear-algebraic motivation for matrix work…

Beans – there’s nothing you’re not seeing, it’s a cockup. The question’s lifted from the predecessor to C1, which was a calculator paper. (There are a few “challenging” non-calculator questions in the textbook too.)

Thanks dr rick. That is a relief! The matrix work is definitely another battle – I more or less threw the book away for that one. I can’t think why it was ever considered adequate for students at this level.

The really astounding thing on the matrix work is that if you trust the board-endorsed book, very occasionally you get hit with an exam question that requires geometric interpretations, either of a diagonalised form of a 2×2 matrix or in terms of interpretations of eigenvectors. This strikes me as profoundly unfair – fortunately I noticed it on doing past paper analysis before the first time I taught it.

I completely agree that it’s a horrible and conceptually barren way to go about matrix work.

What the students tend to find hardest in (current) FP3 is the loci and transformations of complex numbers material – which is fair enough, as doing these questions well requires a good trick bag and extensive practice to develop intuion. On the other hand, all but the strongest also find the vectors work mystifyingly difficult.

EHK – do you teach locally (to Manchester)? (If you’d rather answer that off-line, which I can quite understand, I’m sure Sasha will give you my email address.)